There is considerable interest in retrieving profiles of aerosol effective
radius, total number concentration, and complex refractive index from lidar
measurements of extinction and backscatter at several wavelengths. The
combination of three backscatter channels plus two extinction channels
(3

Here we develop a deeper understanding of the information content and
sensitivities of the 3

The sensitivity metrics allow for identifying (1) information content of the
measurements vs. a priori information; (2) error bars on the retrieved
parameters; and (3) potential sources of cross-talk or “compensating”
errors wherein different retrieval parameters are not independently captured
by the measurements. The results suggest that the 3

Aerosol effects on global and regional climate and human health depend on
aerosol amount, vertical distribution, and proximity to clouds, as well as
the composition, size and absorption properties of the aerosol. The NASA
Aerosol/Clouds/Ecosystems (ACE) mission
(

To achieve these goals, ACE is planned to include a multi-wavelength high
spectral resolution lidar (HSRL) and a multi-wavelength, multi-angle imaging
polarimeter from which vertically resolved aerosol microphysical retrievals
will be made. While passive polarimeter measurements can provide accurate
retrievals of column averaged microphysical properties (Dubovik et al.,
2011; Hasekamp et al., 2011), only lidar measurements can provide the
vertical resolution required. The combination of three backscatter and two
extinction wavelengths (3

There exist various aerosol microphysics retrievals based on 3

The inversion with regularization retrieval (Müller et al., 1999;
Veselovskii et al., 2002) is the standard algorithm used for 3

The linear estimation method (Veselovskii et al., 2012) solves for the particle size distribution represented as a linear combination of the measurement kernels. Only the total integrated number concentration is retrieved rather than the full size distribution. The refractive index is retrieved by iteration, solving the equation for an assumed refractive index and minimizing the resulting systematic error. The systematic error to be minimized is estimated by using only four of the measurements to attempt to reproduce the fifth and repeating for all five measurements. Like the inversion with regularization technique, the final solution is an average of a family of individual solutions.

The arrange and average method (Chemyakin et al., 2014) is a simplified
version of the 3

While it has been demonstrated that 3

Previous studies of the 3

These challenges have been acknowledged and addressed in the existing
retrievals, but there is still relatively little published discussion about
the true sensitivities of the 3

An ideal framework for a study of retrieval sensitivity and information
content is optimal estimation (OE). OE, based on Bayesian statistics, is a
formalized framework for combining measurements, measurement errors,
external information, and constraints. Thoroughly described by Rodgers (2000), it provides a number of key tools for characterizing the
sensitivities and information content of a retrieval system. For example,
Knobelspiesse et al. (2012) use the Shannon information content and the
propagated retrieval errors to characterize the capabilities of multi-angle,
multi-wavelength polarimeter for aerosol microphysics retrieval. Xu and Wang (2015)
analyze the information content of AERONET measurements with respect
to aerosol microphysics retrievals using the propagated retrieval errors and
degrees of freedom (DOF) of the signal. Veselovskii et al. (2005) also
discuss an assessment of the information content and retrieval uncertainties
of the 3

In other words, the diagnostics for sensitivities and information content in the OE
framework do not depend on completing a retrieval. Rather, they depend only
on retrieval inputs: the forward model, measurement uncertainties, and the a
priori constraints. Therefore, although the lidar retrieval algorithms
described above are not OE algorithms, these tools can nevertheless be
usefully applied to this problem to provide implementation-independent
best-case sensitivity metrics. Unlike a perturbation method, the strategy of
performing the sensitivity study using only the forward model allows for
mapping out the entire state space relatively quickly, without the need for
time-consuming retrievals. In addition, since the OE method is a matrix
method, the measurement covariance matrix is handled as a single object,
taking into account measurement errors in all channels simultaneously,
without requiring simplifying assumptions such as an additive property
(Pérez-Ramírez et al., 2013). Finally, the OE method provides a
formalized means of representing the retrieval constraints, a critical part
of an underdetermined retrieval like this, but one which is not well
represented using a perturbation sensitivity study or the eigenvalue
approach of Veselovskii et al. (2005) and Twomey (1977). In this study, we
use an LUT approach to simplify the forward model and set the stage
for a retrieval-independent study of sensitivity and information content of
the 3

In Sect. 2 we describe the overall methodology for our sensitivity study
and in Sect. 3 we describe the specific cases used for illustration in
this paper. In Sect. 4 we give a brief demonstration of the sensitivity of
the 3

With this study, we wish to develop a deeper understanding of the
information content and sensitivities of the 3

The measurements for these retrievals are bulk aerosol extinction and
backscatter coefficients measured by an HSRL or Raman lidar system. They are
related to the particle size distribution and complex refractive index of
the volume of aerosols by this general relationship:

Equation (1) is of the following general form:

If the forward model is linear or can be linearized, then Eq. (2) can be
written by the following matrix equation:

Rodgers (2000) describes the generalized inverse problem, the OE methodology for solving it, and also a number of useful
diagnostics for assessing the information content and retrieval errors.
Although the existing lidar aerosol microphysical retrievals solve the
generalized inverse problem in various ways not limited to OE, the metrics described by Rodgers (2000) are useful for the
retrieval-free information assessment in this project. These include the
scalar DOF metric and the state error covariance
matrix, propagated from the measurement errors. To calculate these metrics,
it is necessary to have the weighting function matrix or Jacobian matrix,

To generate a Jacobian matrix for the purpose of the sensitivity study, we
first simplify the problem by assuming single scattering processes from
spherical particles, monomodal log-normal size distributions, and
wavelength-independent refractive indices. The assumption of
wavelength-independent refractive indices has been used in all 3

Consistent with the assumption of spherical particles and single scattering
processes, we use Mie kernels, which are calculated with code from Bohren
and Huffman (1983). The size distributions are represented as monomodal
log-normal size distributions characterized by the total number
concentration,

In all, five state parameters are used in this study: the median radius and
geometric standard deviation of the monomodal log-normal size distribution,
the total number concentration, and the complex refractive index (real and
imaginary parts). From these, the extinction and 180

For our purposes, the Jacobian matrix is calculated from the LUT using finite differences, using the increments of the LUT itself. The use of finite differences amounts to an assumption that the increments are small enough that the derivatives are locally linear. Testing with both smaller and larger increments confirms that the derivatives are insensitive to the size of the increments from about one-tenth the size of the increments used to at least about 5 times the size used. However, the derivatives and associated retrieval sensitivities are not constant across the entire state space. Therefore, the Jacobians and the metrics describing information content and error propagation have been calculated for several specific realistic cases and also over multiple continuous slices of the hypercube defined by the five state variables, to develop a sense of how these metrics vary over the state space.

Although the published aerosol microphysical retrievals referenced in the
introduction solve the inverse problem in various ways, the LUT can be
thought of as a generalized realization of the forward function, given the
simplifications described above. Since the calculation of the sensitivity
and error metrics (Rodgers, 2000) depend on the forward function but not on
any explicit retrieval, the LUT can be used to assess the 3

Besides the Jacobian matrix, the sensitivity calculations also require the measurement covariance matrix, which depends on the observation system. We use a simple but realistic matrix to describe the measurement errors for this study, modeling the uncertainties as constant, normally distributed relative values with standard deviation of nominally 20 % for the extinction coefficients and 5 % for the backscatter coefficients, and with no correlations between the uncertainties in each channel. Zero or near-zero correlation for the uncertainties between channels is realistic for lidar, for which uncertainties are primarily from random processes (e.g., shot noise) and channel-specific systematic sources (e.g., uncertainty in the filter transmittance). The uncertainty levels used in this study are chosen as realistic targets for a space-based lidar system, based on existing HSRL-2 technology (Hair et al., 2008; Burton et al., 2015). Later in this study (Sect. 7), we explore a few other benchmark values of measurement uncertainties. In reality, uncertainty will not be constant for all aerosol scenarios, but for the purpose of this study, a few benchmark values are sufficient to explore the sensitivities.

State variables and selected derived variables for five constructed reference cases.

The third input needed for these calculations is the a priori covariance matrix. This matrix represents the uncertainty of the prior knowledge of the state. The diagonal terms represent the variance and are chosen so that the standard deviation is represented as one half of the full range in the LUT for each state variable. The off-diagonal terms represent the correlation or covariance between state variables; we assume zero correlation in the a priori. These large prior variances and zero correlations are an intentionally conservative choice. For an actual retrieval, prior information about aerosol type and real aerosol variability would typically be used to decrease these prior variance terms, which can certainly decrease the uncertainty in the final result. Likewise, if it were known a priori that the state variables were correlated, this could also be used to decrease the uncertainty in the final result. However, since our aim is primarily to assess the information content of the measurements themselves, we use conservative prior variance and covariance values for the sensitivity study. We recognize that the state variables are not normally distributed in reality, although the OE formalism makes the assumption that they are (and that the measurement errors likewise are normally distributed). A more advanced strategy would be to use the Markov Chain Monte Carlo method (Posselt and Mace, 2014), which allows for generalized error distributions. However, for this initial study, we use the more straightforward OE method and partially compensate by choosing conservatively large prior variances values.

In describing the calculation of the metrics, we will illustrate the
procedures and interpretation using five particular sets of values in the
state space, which we collectively call “the reference cases.” The values
of the state variables for the five reference cases are given in Table 1, as
well as values of effective radius, effective variance, single scattering
albedo (SSA), and lidar ratio, which are calculated from the state
variables. For a log-normal distribution, the effective radius and effective
variance can be expressed as analytical functions of

The SSA is calculated from the state variables using Mie theory.

The first of the references cases has been constructed to approximately
reflect a real measurement scenario encountered during the DOE TCAP
(Two-Column Aerosol Project) field mission by HSRL-2 (Berg et al., 2015;
Müller et al., 2014); the parameters model a plume of urban outflow. The
complex refractive index for this constructed case is 1.47–0.00325i,
corresponding to a very weakly absorbing aerosol with SSA value of 0.98 at
532 nm. The aerosol is composed of accumulation-mode particles; the
constructed monomodal size distribution has effective radius of 0.17

For the other four reference cases, we vary the state variables in sets.
Cases 2 and 3 have the same complex refractive index as Case 1, but
different size distributions. For Case 2, the effective radius and effective
variance are somewhat larger at 0.24

Case 4 has a size distribution equal to Case 1, but larger real and imaginary refractive index values of 1.61 and 0.03, respectively. For this size distribution, this complex refractive index corresponds to a 532 nm SSA value of 0.89. This can be thought of as similar to a biomass burning plume.

Case 5 is similar to Case 4 in everything except total number concentration.
Now the total number concentration has been increased dramatically to 20 000 cm

These five cases will be used for illustrating the results of the sensitivity analysis, starting in Sect. 5.

The sensitivity of the extinction Ångström exponent to the
effective radius and complex refractive index is shown. The left panel shows
the dependence of extinction Ångström exponent (

First, to build an intuition of the information content encoded within the
3

Recall that aerosol intensive variables are those that do not scale with the amount of aerosol loading. Of the five state variables, total number concentration is an extensive variable while the other four (real and imaginary parts of the refractive index, median radius, and geometric standard deviation) are intensive variables. Aerosol extinction and backscatter coefficients, the direct measurements of a lidar using the HSRL or Raman techniques, are extensive variables; ratios of these basic measurements are intensive variables. Burton et al. (2012) show that intensive variables such as the lidar ratio (extinction-to-backscatter ratio at a given wavelength) and backscatter color ratio (i.e., ratio of backscatter at two different wavelengths) encode information about the type of aerosol present in broad categories, i.e., marine vs. smoke vs. urban pollution. It is also known that the extinction Ångström exponent is sensitive to the particle size distribution (e.g., Schuster et al., 2006; Ångström, 1929; Kaufman et al., 1994).

Like Fig. 1 but for 532 nm lidar ratio (top row) and 355 nm lidar ratio (bottom row).

Like Fig. 1 but for the backscatter color ratio (which is the ratio of the aerosol backscatter coefficient at 532 nm divided by the aerosol backscatter coefficient at 1064 nm).

Figure 1 (left panel) illustrates the monotonic dependence of extinction
Ångström exponent (355/532 nm) on the effective radius, for
monomodal log-normal size distributions. The sensitivity of this parameter
to either the real or imaginary part of the refractive index is smaller, as
demonstrated by shallower slopes in the middle and right panels of Fig. 1.
However, Fig. 2 illustrates the dependencies of the lidar ratio
(at 532 and 355 nm), which is the ratio of the extinction to backscatter
and is also the inverse of the product of the aerosol 180

The degrees of freedom (DOF) of the signal,

While this simple sensitivity check illustrates that changes in the aerosol microphysical parameters are reflected in the measurements, it is not sufficient to determine if the measurements are enough to retrieve all five state parameters. For that, we must turn to more quantitative tools.

The retrieval problem as specified above consists of five direct aerosol
measurements (two extinction and three backscatter measurements) from a
lidar system at a single level in the atmosphere and five state vector
elements (three describing the number and size distribution and two to
specify the complex refractive index). We would like to know if the five
measurements are sufficient to determine the five unknowns – in other words,
to determine if the inverse system is fully determined, overdetermined, or
underdetermined and by how much. Rodgers (2000) describes a useful metric to
quantify the number of pieces of independent information in the measurement,
the DOF for the signal,

Since the error-normalized Jacobian matrix is weighted by the prior
covariance in the numerator and the measurement error in the denominator,
elements greater than unity indicate where variability in the true state
exceeds measurement noise. The trace of the matrix,

For the first reference case, Case 1 (see Table 1 for description), the
signal DOF,

Propagated state error covariance matrix for the first reference case, assuming measurement errors of 5 % for backscatter and 20 % for extinction and a priori covariance as described in Sect. 6 and Table 3.

Propagated uncertainties (standard deviations) for state variables and selected additional variables derived from the state variables, shown for the reference cases described in Table 1. The uncertainties are shown as absolute value for all variables with relative uncertainty in parenthesis for the size distribution variables. The propagated uncertainties (Eq. 9) depend on assumed measurement errors of 5 % for backscatter and 20 % for extinction and depend on a priori covariance as described in the text. The assumed a priori uncertainty and the requirements described in the ACE white paper (also 1 standard deviation) are listed for comparison.

The quoted

Signal values for the DOF less than 5 mean that some of the
information in the five retrieved parameters is not provided directly by the
measurements and will be “filled in” by a priori information or other
constraints in a retrieval. A value of

While the signal DOF is a useful metric that indicates the number of
independent pieces of information in the measurements with respect to the
state, the a posteriori (i.e., propagated) state error covariance matrix is
more useful both for indicating how the retrieval errors are propagated from
the measurement errors and also for assessing how the under-determinedness
affects specific state variables. The state error covariance matrix,

Table 2 shows

Since the prior covariance matrix was defined rather conservatively in this
study, the reduction from the prior uncertainty may be less useful than
comparing to uncertainty values defined in terms of a desired goal. Part of
the motivation of this study is to determine the extent to which a 3

Some of the ACE requirements are stated in terms of the effective radius,
effective variance, and SSA, quantities that are not part of the nominal set
of state variables described above; however, they are directly related to
the state variables and can be derived from them. In general, if a secondary
variable,

For our purpose, the variable

The effective radius and effective variance can be calculated for a
monomodal log-normal size distribution using Eqs. (

As with the signal DOF, the propagated errors for the state vector elements and for effective radius, effective variance, and SSA are regime dependent, varying over different parts of the state space. In the Supplement, there are figures similar to Fig. S4 but which show five state variables as well as the derived variables, effective radius, effective variance, and SSA. These illustrate the ease with which the sensitivity metrics can be calculated for the whole state space, but since some of the states represented in these slices may not be particularly realistic, it can be hard to interpret the results. Therefore, the five reference cases in Table 1 were designed to provide a focus for understanding the regime dependence more easily.

Recall that the differences between cases 1, 2, and 3 are related to the size distribution. The size distribution for Case 3 is a coarse mode with a larger particle size, larger geometric standard deviation, and smaller total number concentration than cases 1 and 2. Compared to Case 1 or 2, Case 3 has larger propagated relative uncertainty of the effective radius, 50 % uncertainty compared to 23 and 29 %, and also of total number concentration, 122 % compared to 98 and 103 %, mostly due to the increase in the geometric standard deviation. For the most part, we found increasing relative uncertainties for the size distribution parameters for increasing geometric standard deviations (with some exceptions, as can be seen in the figures in the Supplement). However, compared to Case 1, Case 3 has smaller uncertainties on the complex refractive index and SSA, although the complex refractive index did not change between cases.

Case 4 has the same size distribution as Case 1, but the complex refractive index corresponds to a more absorbing aerosol. There are only minor differences in the size distribution uncertainties, but the uncertainties on the complex refractive index and SSA increase, suggesting less sensitivity in the retrieval to the complex refractive index of absorbing aerosols.

Case 5 is identical to Case 4 except that it has a very large total number concentration. Although such a large total number concentration in a real-world measurement scenario would mean greatly increased signal-to-noise ratio (SNR), the measurement errors for this study are defined as constant percentages, so the SNR effect is not reflected in this study. Instead, total number concentration behaves essentially as a scaling variable in the retrieval, and therefore most of the propagated uncertainties are very similar for Case 5 compared to Case 4. The exception is the uncertainty in the total number concentration itself, which decreases from 94 to 68 %.

Propagated uncertainties for Case 1, expressed as absolute values and percentage (for the size distribution parameters) for three different theoretical instrument configurations with different backscatter and extinction uncertainties. The last column repeats the draft requirements from the ACE white paper as in Table 3 for reference.

Comparing the propagated uncertainties to the ACE requirements, note that
ACE calls for an uncertainty on the column total number concentration of
100 %. The uncertainties in Table 3 show that the 3

The uncertainties on the vertically resolved effective variance are 36 and 41 % for the absorbing cases, which already meets the proposed ACE column requirement of 50 %. The non-absorbing fine-mode and coarse-mode cases have effective variance uncertainties of 61 to 68 %, not very much larger than the ACE column requirement.

The requirement of 10 % column uncertainty for the effective radius is not met for any of the five illustrated cases on a vertically resolved basis; the propagated uncertainties are 2 to 3 times larger for the three fine-mode cases and 5 times larger for the coarse-mode case. A factor of 2 or 3 may be recoverable by a profile retrieval which uses multiple vertically resolved measurement levels simultaneously, assuming the aerosol properties are correlated across several levels.

The propagated uncertainty on the real refractive index is 2 to 7 times the proposed ACE column requirement, in this case smallest for the coarse-mode case and worst for the two absorbing fine-mode cases.

The proposed ACE requirement for SSA is 0.02 on a vertically resolved basis. The propagated uncertainties for all four cases are 3 to 5 times larger than this proposed requirement, which may be sufficient for distinguishing extreme cases such as intense biomass burning plumes and also may be reducible to some extent by a profile retrieval.

State correlation matrix derived from the covariance matrix shown in Table 2, showing the correlations between retrieved variables for Case 1, assuming measurement errors of 5 % for backscatter and 20 % for extinction and a priori uncertainties from Table 3.

Correlation matrix of the retrieved variables for Case 2, assuming the same measurement errors of 5 % for backscatter and 20 % for extinction and a priori uncertainties listed in Table 3.

It should perhaps be mentioned that the ACE requirements are not necessarily finalized and the values quoted here are draft requirements. Similarly, the instrument performance used for the results described above is only approximate based on a best-guess estimate of realistic targets for a space-based lidar system, based on the technology used for the airborne HSRL-2. Since the motivation for this study is to determine what retrieval performance is possible from a lidar-only microphysical retrieval, it is useful to briefly explore the retrieval uncertainties as a function of instrument performance. Table 4 accordingly shows the propagated uncertainties, using reference Case 1, for three different instrument configurations with different measurement uncertainties for backscatter and extinction. The first measurement configuration assumes that the uncertainties are larger than previously described, 10 % for aerosol backscatter and 30 % for aerosol extinction. The second of the three configurations in Table 4 is a repetition from Table 3, with uncertainties of 5 and 20 % on aerosol backscatter and extinction, respectively. The third theoretical instrument configuration is more ambitious, with assumed uncertainty 5 % on aerosol backscatter and 10 % on aerosol extinction. Comparing the first and second scenarios, when the measurement uncertainties are allowed to increase as described, the retrieval uncertainties increase by a factor of 20–50 %. Comparing the second and third scenarios, when instead the extinction measurement uncertainty is decreased by half, then the retrieval uncertainties all decrease by approximately 30–40 %. In the third scenario, the draft ACE requirement for vertically resolved total number concentration is met; the requirement for column effective variance is met even on a vertically resolved basis, and the vertically resolved effective radius uncertainty is less than twice the column requirement. However, the real refractive index and SSA uncertainties are still large compared to the ACE draft requirements. This level of precision and accuracy may be difficult to achieve with a satellite lidar.

Recall that the proposed ACE system consists of both a multi-wavelength HSRL and a multi-wavelength, multi-angle polarimeter. The current sensitivity study addresses only the lidar. A combined retrieval with both lidar and polarimeter will certainly have higher information content particularly pertaining to aerosol absorption, and a better chance of meeting all of the draft ACE requirements. To quantitatively assess the information content of this more complicated system, a full column retrieval using a combined lidar-plus-polarimeter forward model would be required, which is outside of the scope of the current paper.

Based on the current study, it seems likely that a 3

Besides the diagonal variance elements, the state error covariance matrix
includes off-diagonal terms that describe the interaction between pairs of
state variables in the retrieval. Prior similar sensitivity studies for
other systems do not explicitly address the off-diagonal terms of the
propagated matrix (Xu and Wang, 2015; Knobelspiesse et al., 2012), but these
terms give critical information about retrieval performance. To illustrate,
Tables 5 and 6 give the state error correlation matrix for Case 1 and
Case 2, respectively. These can be easily converted from the state error
covariance matrices, like the one given in Table 2 for Case 1. The
correlation matrices show that there is some correlation between all pairs
of variables, with the highest correlations between the real and imaginary
parts of the complex refractive index and between the total number
concentration and median radius. The correlations have a complicated regime
dependence, illustrated in Figs. 5, S9, and S10. Although cases 1 and 2 vary
only a small amount in median radius and geometric standard deviation, there
is a significant increase for Case 2 in the magnitude of the correlations
between size distribution variables. For Case 2, the correlation is

The a posteriori correlation between retrieved total number
concentration and median radius is here shown as a 2-D slice through the
five-variable state space. The complex refractive index is held fixed at
1.470–0.00325i, the total number concentration is held fixed at 1001 cm

High magnitude correlations between the retrieved variables indicate the potential for cross-talk between these parameters. Cross-talk can cause additional error in the retrieved parameters that is not reflected in the variance terms, due to non-unique solutions which have compensating errors. In an ideal case with no cross-talk, the forward model evaluated at the true state would produce output equal to the measurements (ignoring measurement error), while the forward model evaluated using an incorrect state vector should produce output that does not agree with measurements. However, in the case of compensating errors or cross-talk, an incorrect solution may also reproduce the measurements if, for example, an error in the median radius that tends to produce larger backscatter and extinction values were compensated by an error in the total number concentration that tends to produce smaller values. Such compensating errors make it impossible for the measurements to distinguish between the true state and the erroneous state.

The cause of the cross-talk can broadly be described as a lack of sensitivity in the measurements. The cross-talk between total number concentration and median radius occurs because particles significantly smaller than the shortest measured wavelength (355 nm) contribute little to observed optical properties. Therefore, the measurements can be insensitive to the difference between large numbers of very small particles and smaller numbers of larger particles. This problem and a partial remediation are examined in more detail in Sect. 9. The cross-talk between the real and imaginary index of refraction is related to a relative lack of sensitivity to absorption in the lidar measurements. Probably the best remedy for this latter problem is to incorporate additional information content into the retrieval, preferably in the form of additional coincident measurements, as from a polarimeter on the same platform.

Taking measurement error into account, there are always multiple solutions that reproduce the measurements to within the measurement error. This is not a concern when the solutions are clustered around the true solution, but it can be a significant issue in the case of cross-talk or compensating errors as discussed above. Figures 6 and 7 show histograms of the number of solution states in the gridded LUT that reproduce the backscatter and extinction values of Case 1 to within the prescribed error bars (5 % for backscatter coefficient and 20 % for extinction coefficient). Figure 6 shows the total number concentration and Fig. 7 shows the median radius, respectively. Note that although the peaks of the histograms do not exactly match the specified values for Case 1 (indicated by dashed lines), the solutions are clustered around those values.

Histograms showing the total number concentration value for all solutions in the gridded LUT (i.e., without interpolation) that match the backscatter and extinction coefficients of Case 1 within measurement errors of 5 % for backscatter and 20 % for extinction. The total number concentration value for Case 1 is marked with a dashed line. Red histogram bars show solutions from the full gridded LUT. Blue histogram bars show solutions from the modified LUT, which excludes size distributions that have a significant contribution from particles of smaller than 50 nm radius.

Histograms showing the median radius for all solutions in the gridded LUT that match the backscatter and extinction coefficients of Case 1 within measurement errors of 5 % for backscatter and 20 % for extinction. The dashed line indicates the median radius for Case 1. Red and blue are as in Fig. 6.

Like Fig. 6 but for Case 2.

Like Fig. 7 but for Case 2. The inset box shows the blue
histograms (reduced solution set) with an expanded

In contrast, Figs. 8 and 9 illustrate the set of solutions from the gridded LUT that match Case 2 within the measurement error bars (shown in red). Figure 8 shows that this set of solutions covers an enormous range in total number concentration. The range of total number concentration for these solutions is much larger than indicated by the propagated standard deviation shown in Table 6. Cases 1 and 2 have similar propagated standard deviations; the problem with Case 2 is only evident in the near-unity correlation value between total number concentration and median radius, shown in Table 6. The very high correlation indicates that the solutions with very large total number concentrations are those solutions that also have very small median radii. Very small particles contribute little to extinction or backscatter at the lidar wavelengths. So, large numbers of very small particles can be included in the retrieved solution without significantly affecting the agreement with the measurements; therefore, the measurements by themselves are not sufficient to determine if these very small particles are actually present.

This situation emphasizes the value of examining the cross-terms of the propagated error matrix. The regime dependence of this situation is complex and the problem can be detected only by studying the correlation matrix or else by examining the distribution of solutions for a given retrieval.

A resolution of the cross-talk can be achieved by adding an additional
constraint on either the total number concentration or the radius using a
priori information. For example, it is probably unrealistic to allow total
number concentrations up to 40 000 cm

A radius of 50 nm is proposed for the cutoff value based on the sensitivity of
the lidar measurements and on the naturally occurring lower bound of the
atmospheric aerosol accumulation mode. Typically aerosol size distributions
are described in terms of three or four size modes, depending on whether one is
examining the number- or mass-based size distribution (Seinfeld and Pandis,
2006). Most particles, on a number basis, exist in the ultrafine diameter
size range of a few nanometers up to a few hundred nanometers, with two
distinct modes: the nucleation mode (

Clearly, if an Aitken or nucleation mode with large number concentration does exist, limiting the size range of the retrieval introduces the possibility of bias in total number concentration. However, it is important to realize that even if it is known from external sources (such as in situ measurements) that an observation is occurring in a region of significant new particle production, lowering the cutoff radius will not resolve the systematic error in the retrieval, since the measurements cannot distinguish between large numbers of very small particles and smaller numbers of larger particles. Therefore, we think it is sensible to limit the particle size in the retrievals to reflect the measurement sensitivity to larger sized particles. This strategy also has the benefits of making the constraint explicit and leading to a clear and understandable interpretation of the results. In this case, the retrieval should not be described as a retrieval of total aerosol number concentration but rather as a retrieval of accumulation-mode and coarse-mode aerosols, more accurately reflecting the retrieval sensitivities.

This strategy has heritage in existing retrievals. In inversion with regularization (Müller et al., 1999), the under-determination of the retrieval is addressed by putting strong constraints on the window of particle sizes that are considered, effectively limiting the minimum particle radius to 50 nm (Veselovskii et al., 2002). However, in that retrieval the limit varies from case to case and even from one solution to another within the set of solutions that are averaged for a particular retrieval, with the minimum radius being anything between 50 and 500 nm. Since we argue that the need for a minimum particle radius cutoff is related to the limited sensitivity of the measurements to very small particles, we believe that a single cutoff would be more consistent with our understanding of the retrieval sensitivities. In any case, it is important to recognize that the size cutoff amounts to prior information supplementing the information content of the measurements; explicitly describing the prior information is essential to understanding and evaluating retrieval systems and their products.

The effect on the backscatter coefficient (solid lines) and
extinction coefficient (dashed lines) of a narrow Aitken mode with median
radius

To investigate the potential for bias associated with the particle size
cutoff, it is useful to examine how much of an effect the Aitken mode would
have on the backscatter and extinction measurements. For this exercise, we
start with a retrieval case similar to an actual measurement described by Müller et al. (2014) from the NASA
Langley HSRL-2 on 17 July 2012, from TCAP, and then add on a simulated mode with
particle radius of 15 nm (diameter

The effect on the backscatter coefficient (solid lines) and
extinction coefficient (dashed lines) of 1000 cm

As the particle radius gets larger, the sensitivity of the measurements to
these aerosols increases. Figure 11 shows the effect as a fraction of
backscatter and extinction (again using the measurements from the TCAP case
on 17 July 2012 as a reference) of 1000 cm

It is worth pointing out that although it is true that lidar measurements lack sensitivity to particles much smaller than the smallest wavelength, they do not lack sensitivity to particles much larger than the longest wavelength, as is sometimes stated. For instance, it is not true that “pollens cannot be observed with lidar systems” (Bockmann et al., 2005). See Fig. 12 for an illustration of lidar measurements simulated by Mie modeling for very large particles. At these large particle sizes, a forward model for the lidar based only on the single scattering Mie calculations is no longer applicable, but this simple illustration serves to show that the backscatter and extinction coefficients are much larger, not smaller, than the benchmark observations of the lidar. The scattering efficiency of large particles is significant even at wavelengths much smaller than the particle size and so the effect of laser light scattering from large particles is easily seen using lidar. However, since the particle size dependence of the lidar measurements is not monotonic at large particle sizes and the single scattering forward model is no longer applicable, microphysical retrievals of particle properties are challenged at large particle sizes. See Gasteiger and Freudenthaler (2014) for a further discussion of retrieval of large particle size from multi-wavelength lidar.

The effect on the backscatter (red line) and extinction (blue
line) coefficients at 532 nm of a narrow mode (geometric standard deviation
1.1) of large particles (radius varies along

The a posteriori correlation between retrieved total number
concentration and median radius is here, as a function of median radius and
geometric standard deviation, with the complex refractive index held fixed
at 1.470–0.00325i and the total number concentration held fixed at 1001 cm

Like Table 3, but using total volume concentration instead of total number concentration as a state variable, this table shows propagated uncertainties (standard deviations) for state variables and selected additional variables derived from the state variables, shown for the reference cases described in Table 1. The propagated uncertainties (Eq. 9), depend on assumed measurement errors of 5 % for backscatter and 20 % for extinction and depend on a priori covariance as described in the text. The assumed a priori uncertainties are listed for comparison.

It is known from, for example, Veselovskii et al. (2004) that performing the
retrieval with higher-order kernels may reduce the retrieved uncertainties.
It is straightforward to use the volume size distribution instead of the
number size distribution for

There is considerable interest in retrievals of aerosol size distribution
parameters and absorption properties using multi-wavelength HSRL or Raman lidar. While there have been successful 3

We find that the five 3

In general, information about the state vector that is not provided by the measurements comes from assumptions, constraints, or other a priori information. Smoothing and regularization are examples of retrieval constraints, as is the idea of limiting the minimum particle radius. Retrieval constraints and assumptions can also be hidden or difficult to characterize. For specific retrieval methodologies, we would like to emphasize the importance of explicitly describing any prior information and constraints that affect retrieval results.

In this sensitivity study, only very conservative constraints were used in
order to pinpoint the sensitivity of the measurements. To achieve better
performance with a retrieval, three strategies can be adopted either singly
or in combination:

Add a priori information that constrains the retrieval using known information about the observed aerosol.

Reduce the measurement uncertainties.

Add additional measurements to the system.

One method to assign a priori covariance information is to use aerosol classification from the lidar intensive parameters (Burton et al., 2012) to infer what type of aerosol is present and then assign prior variances for the state parameters that are specific to that aerosol type. It has been demonstrated that the lidar intensive parameters from an HSRL have sufficient information content to categorize aerosol into broad categories. Assigning a priori values based on these categories additionally requires representative information about the microphysical properties of aerosols in each category from in situ measurements or from modeling.

Reducing the measurement uncertainty involves either designing the observing
system to stricter requirements (to the extent practical) or reworking the
retrieval problem to make more optimal use of the measurement information.
For example, a simultaneous profile retrieval that uses the 3

Finally, measurement information content can be increased by adding more measurements to the system, for example by combining coincident lidar plus polarimeter measurements from the same platform. This combination is expected to add significantly more information content and reduce the need for constraints or a priori information

Research is ongoing into each of the three retrieval strategies described above, aerosol-type-specific prior covariance matrices, profile retrievals, and combined lidar plus polarimeter retrievals. Additional sensitivity studies for these scenarios will be performed in the future.

Funding for this research came from the NASA Aerosol/Clouds/Ecosystems project, NASA Earth Science Division's Remote Sensing Theory program, and NASA Radiation Science Program. Edited by: V. Amiridis Reviewed by: two anonymous referees